Group #15672104 — subgroups

Subgroups — #15672104

Order 2944.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092972	C₂	1	1	yes	maximal, characteristic
2	#13092473	C₂	1	1	yes	maximal, characteristic
3	#13092469	C₂	1	1	yes	maximal, characteristic
4	#13092165	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092165	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#10625117	—	2	1	no	—
10	Q₈×Dic₂₃	—	2	1	no	—
11	#10625068	C₄	1	2	yes	—
12	#10625063	—	2	1	no	—
13	#10625061	—	2	1	no	—
14	#10625058	C₄	1	2	yes	—
15	C₄²×D₂₃	C₂²	1	1	yes	characteristic
16	#10625040	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	(C₂×D₂₃)⋊C₈	—	2	1	no	—
20	(C₂×D₂₃)⋊C₈	—	2	1	no	—
21	#10624997	—	2	1	no	—
22	C₈×Dic₂₃	—	2	1	no	—
23	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₂₃	—	2	1	no	—
26	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
28	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
29	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₂₃	—	2	1	no	—
31	C₂×C₈×C₂₃	—	2	1	no	—
32	C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	(C₂×D₂₃)⋊C₄	—	2	2	no	—
36	C₄⋊Dic₂₃	—	2	1	no	—
37	C₄⋊Dic₂₃	—	2	1	no	—
38	C₄⋊Dic₂₃	—	2	2	no	—
39	Dic₂₃⋊C₄	—	2	2	no	—
40	C₄×Dic₂₃	—	2	2	no	—
41	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₂₃	D₄	1	1	yes	characteristic
43	C₄×Dic₂₃	D₄	1	1	yes	characteristic
44	C₂×C₂₃⋊C₈	—	2	1	no	—
45	C₂×C₂₃⋊C₈	—	2	1	no	—
46	C₂²×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	Q₈×C₂₃	—	4	1	no	—
48	Q₈×C₂₃	—	4	1	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×D₂₃	—	2	1	no	—
58	C₄×D₂₃	—	2	1	no	—
59	C₄×D₂₃	—	2	1	no	—
60	C₄×D₂₃	—	2	1	no	—
61	C₄×D₂₃	—	4	1	no	—
62	Dic₄₆	—	4	2	no	—
63	C₈×C₂₃	—	4	1	no	—
64	C₈×C₂₃	—	4	1	no	—
65	C₂₃⋊C₈	—	4	1	no	—
66	C₂₃⋊C₈	—	4	1	no	—
67	#1554	—	23	1	no	maximal, Sylow
68	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×D₂₃	—	2	1	no	—
70	C₂×D₂₃	—	2	1	no	—
71	C₂×D₂₃	—	2	1	no	—
72	C₄×C₂₃	—	2	1	no	—
73	C₄×C₂₃	—	4	2	no	—
74	C₄×C₂₃	#135	1	1	yes	characteristic
75	C₄×C₂₃	#135	1	1	yes	characteristic
76	C₄×C₂₃	#137	1	1	yes	characteristic
77	C₄×C₂₃	#137	1	1	yes	characteristic
78	Dic₂₃	—	2	1	no	—
79	Dic₂₃	—	2	1	no	—
80	Dic₂₃	—	2	1	no	—
81	Dic₂₃	—	4	2	no	—
82	#533	—	23	1	no	—
83	#432	—	23	1	no	—
84	#431	—	23	1	no	—
85	#333	—	23	1	no	—
86	#333	—	23	1	no	—
87	#332	—	23	1	no	—
88	#332	D₂₃	1	1	yes	characteristic, complemented
89	C₂×C₂₃	#413	1	1	yes	characteristic
90	C₂×C₂₃	#416	1	1	yes	characteristic
91	C₂×C₂₃	#419	1	1	yes	characteristic
92	D₂₃	—	4	1	no	—
93	C₄⋊Q₈	—	23	1	no	—
94	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
95	#125	—	46	1	no	—
96	C₂²⋊Q₈	—	23	2	no	—
97	C₂²⋊Q₈	—	46	1	no	—
98	C₄×Q₈	—	23	2	no	—
99	C₄×Q₈	—	46	1	no	—
100	#117	—	46	1	no	—
101	C₂×C₄²	—	23	1	no	—
102	C₄⋊C₈	—	23	1	no	—
103	C₄⋊C₈	—	23	1	no	—
104	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
106	C₂²⋊C₈	—	46	1	no	—
107	C₂²⋊C₈	—	46	1	no	—
108	#97	—	46	1	no	—
109	C₄×C₈	—	46	1	no	—
110	C₂₃	#1554	1	1	yes	characteristic, complemented, Sylow
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	23	2	no	—
113	C₂²×C₄	—	23	1	no	—
114	C₂²×C₄	—	23	1	no	—
115	C₂²×C₄	—	23	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	46	1	no	—
119	C₂×C₈	—	46	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	46	2	no	—
122	C₄⋊C₄	—	46	2	no	—
123	C₄⋊C₄	—	46	1	no	—
124	C₄⋊C₄	—	46	1	no	—
125	C₄⋊C₄	C₄×D₂₃	1	2	yes	—
126	C₂²⋊C₄	—	46	2	no	—
127	C₄²	—	23	1	no	—
128	C₄²	—	23	1	no	—
129	C₄²	—	23	1	no	—
130	C₄²	—	46	2	no	—
131	C₄²	C₂²×D₂₃	1	1	yes	characteristic
132	C₂³	—	23	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	92	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	23	1	no	—
138	C₂×C₄	—	23	1	no	—
139	C₂×C₄	—	23	1	no	—
140	C₂×C₄	—	46	1	no	—
141	C₂×C₄	—	46	1	no	—
142	C₂×C₄	—	46	1	no	—
143	C₂×C₄	—	46	1	no	—
144	C₂×C₄	—	46	2	no	—
145	C₂×C₄	—	92	1	no	—
146	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
147	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
148	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	92	1	no	—
152	C₈	—	92	1	no	—
153	C₂²	—	46	1	no	—
154	C₂²	—	46	1	no	—
155	C₂²	—	46	1	no	—
156	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	46	1	no	—
160	C₄	—	46	1	no	—
161	C₄	—	46	1	no	—
162	C₄	—	92	2	no	—
163	C₄	#10625083	1	1	yes	characteristic
164	C₄	#10625086	1	1	yes	characteristic
165	C₄	#10625087	1	1	yes	characteristic
166	C₄	#10625089	1	1	yes	characteristic
167	C₂	—	92	1	no	—
168	C₂	#13092400	1	1	yes	characteristic
169	C₂	#13092410	1	1	yes	characteristic
170	C₂	#13092424	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092972

C₂

yes

maximal, characteristic

#13092473

C₂

yes

maximal, characteristic

#13092469

C₂

yes

maximal, characteristic

#13092165

C₂

yes

maximal, characteristic, complemented

#13092165

C₂

yes

maximal, characteristic, complemented

#13092163

C₂

yes

maximal, characteristic, complemented

C₂₃×#332

C₂

yes

maximal, characteristic, complemented

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

#10625117

—

Q₈×Dic₂₃

—

#10625068

C₄

yes

—

#10625063

—

#10625061

—

#10625058

C₄

yes

—

C₄²×D₂₃

C₂²

yes

characteristic

#10625040

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

(C₂×D₂₃)⋊C₈

—

(C₂×D₂₃)⋊C₈

—

#10624997

—

C₈×Dic₂₃

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×C₄×D₂₃

C₂×C₄

yes

characteristic

C₂×C₄×D₂₃

D₄

yes

characteristic

C₂×C₄×D₂₃

D₄

yes

characteristic

C₂×Dic₄₆

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

D₄

yes

characteristic

C₄×Dic₂₃

D₄

yes

characteristic

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂²×D₂₃

C₂²⋊C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₃

C₂²⋊C₄

yes

characteristic

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1554

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

D₂₃

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

100

#117

—

101

C₂×C₄²

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₂₃

yes

characteristic

105

C₄⋊C₈

C₂×D₂₃

yes

characteristic

106

C₂²⋊C₈

—

107

C₂²⋊C₈

—

108

#97

—

109

C₄×C₈

—

110

C₂₃

#1554

yes

characteristic, complemented, Sylow

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

C₄×D₂₃

yes

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₂₃

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

147

C₂×C₄

D₄×D₂₃

yes

characteristic

148

C₂×C₄

D₄×D₂₃

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

C₂²⋊C₄×D₂₃

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#10625083

yes

characteristic

164

C₄

#10625086

yes

characteristic

165

C₄

#10625087

yes

characteristic

166

C₄

#10625089

yes

characteristic

167

C₂

—

168

C₂

#13092400

yes

characteristic

169

C₂

#13092410

yes

characteristic

170

C₂

#13092424

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#332	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₃	C₂²⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	#332	D₂₃
hypocenter	C₂₃	#1554
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#332

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₃

C₂²⋊C₄

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #13092165	C₂	semidirect
5. #13092165	C₂	semidirect
6. #13092163	C₂	semidirect
7. C₂₃×#332	C₂	semidirect
88. #332	D₂₃	semidirect
110. C₂₃	#1554	semidirect

H ≅ G/N

type

4. #13092165

C₂

semidirect

5. #13092165

C₂

semidirect

6. #13092163

C₂

semidirect

7. C₂₃×#332

C₂

semidirect

88. #332

D₂₃

semidirect

110. C₂₃

#1554

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13092972	2
#13092473	2
#13092469	2
#13092165	2
#13092163	2
C₂₃×#332	2
#1554	23

subgroup

index

C₂₃×#332

Subgroups — #15672104

Conjugacy classes of subgroups

Complements

Maximal subgroups